 
Summary: On Linear Inference
Frank Pfennnig
Draft of February 2, 2008
Inference. When we write an inference rule
even(x) true
even(s(s(x))) true
what do we mean? To discuss this, some terminology: we say "even(t)" is a
proposition and "even(t) true" is a judgment. Following MartinL¨of, a judgment
is an object of knowledge. We obtain knowledge by making inferences from
judgments we already know. We can then read the rule above as
If we know that t is even for a term t, we may conclude (and thereby
know) that s(s(t)) is also even.
The process of inference is therefore one by which we gain knowledge. We may
start with the knowledge that 0 is even, then we gain the information that
s(s(0)) is even, and next we learn that s(s(s(s(0)))) is even, and so on. This
process is clearly monotonic: we gain knowledge, but we never forget, at least
in the idealized world of mathematics.
Persistent and Ephemeral Truth. It has long been argued by philosophers
that truth is not universal, but depends on the world in which we consider a
proposition. This could depend on time ("It is raining now, but it did not rain
